# Conjugacy-separable implies residually finite

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This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., conjugacy-separable group) must also satisfy the second group property (i.e., residually finite group)
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## Definitions used

### Conjugacy-separable group

Further information: Conjugacy-separable group

A group $G$ is termed conjugacy-separable if given any elements $x,y \in G$ that are not conjugate in $G$, there exists a normal subgroup of finite index $N$ such that the images of $x and [itex]y$ in $G/N$ are not conjugate.

### Residually finite group

Further information: Residually finite group

A group $G$ is termed residually finite if given any non-identity element $x \in G$, there is a normal subgroup of finite index $N$ in $G$ such that $x$ is not in $N$.

## Proof

Given: A conjugacy-separable group $G$, a non-identity element $x \in G$.

To prove: There is a normal subgroup of finite index $N$ in $G$ such that $x$ is not in $N$.

Proof: Let $e$ be the identity element. Since no non-identity element is conjugate to the identity element, $x$ and $e$ are in distinct conjugacy classes. Thus, there exists a normal subgroup $N$ of finite index such that the image of $x$ is not conjugate to the image of $e$. In particular, this implies that $x$ is not contained in $N$, so we are done.